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Full text of "Analytical Mechanics"

IMPULSE AND MOMENTUM                     253
compared with the pile or the nail. On the other hand, for shaping rivets the anvil must have a large mass compared with the hammer.
203. Motion where Moving Mass Varies.  If the moving mass varies, as in the case of an avalanche, the relation between impulse and momentum still holds, that is, impulse equals the change in the momentum. We have to take into account, however, the change in the momentum of the mass which is continually added to the moving system as well as that of the original mass. Let a mass dm be added in the time dt] then supposing dm to have been initially at rest the total change in the momentum is m dv + v dm, where the first term is the increase in the momentum of m and the second term is the increase in the momentum of dm. Therefore the impulse given by the resultant force dF in the time dt is
F dt = m dv -f v dm, .        dv  ,    dm
F=m+
which is the same equation as (III), except that in (III) m was considered to be constant, while here it is considered as a variable.
If dm has an initial velocity u, then the change in the momentum of dm is (v u) dm.    Therefore
dv  ,  f         ^ -+(v-u)
f     N       dm
/-TTTTT\
(xm>